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Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2022 Abelkonkurransen Finale
2022 Abelkonkurransen Finale
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(7)
4b
1
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Do there exist 2022 polynomials of degree 2021?
Do there exist
2022
2022
2022
polynomials with real coefficients, each of degree equal to
2021
2021
2021
, so that the
2021
⋅
2022
+
1
2021 \cdot 2022 + 1
2021
⋅
2022
+
1
coefficients in their product are equal?
4a
1
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Functional inequality involving reciprocals and a square root
Find all functions
f
:
R
+
→
R
+
f:\mathbb R^+ \to \mathbb R^+
f
:
R
+
→
R
+
satisfying \begin{align*} f\left(\frac{1}{x}\right) \geq 1 - \frac{\sqrt{f(x)f\left(\frac{1}{x}\right)}}{x} \geq x^2 f(x), \end{align*} for all positive real numbers
x
x
x
.
3
1
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Smallest number of squares to color on a board
Nils has an
M
×
N
M \times N
M
×
N
board where
M
M
M
and
N
N
N
are positive integers, and a tile shaped as shown below. What is the smallest number of squares that Nils must color, so that it is impossible to place the tile on the board without covering a colored square? The tile can be freely rotated and mirrored, but it must completely cover four squares. [asy] usepackage("tikz"); label("% \begin{tikzpicture} \draw[step=1cm,color=black] (0,0) grid (2,1); \draw[step=1cm,color=black] (1,1) grid (3,2); \fill [yellow] (0,0) rectangle (2,1); \fill [yellow] (1,1) rectangle (3,2); \draw[step=1cm,color=black] (0,0) grid (2,1); \draw[step=1cm,color=black] (1,1) grid (3,2); \end{tikzpicture} "); [/asy]
2b
1
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The ratio between areas of triangles ABC and DEF
Triangles
A
B
C
ABC
A
BC
and
D
E
F
DEF
D
EF
have pairwise parallel sides:
E
F
∥
B
C
,
F
D
∥
C
A
EF \| BC, FD \| CA
EF
∥
BC
,
F
D
∥
C
A
, and
D
E
∥
A
B
DE \| AB
D
E
∥
A
B
. The line
m
A
m_A
m
A
is the reflection of
E
F
EF
EF
through
B
C
BC
BC
, similarly
m
B
m_B
m
B
is the reflection of
F
D
FD
F
D
through
C
A
CA
C
A
, and
m
C
m_C
m
C
the reflection of
D
E
DE
D
E
through
A
B
AB
A
B
. Assume that the lines
m
A
,
m
B
m_A, m_B
m
A
,
m
B
, and
m
C
m_C
m
C
meet in a common point. What is the ratio between the areas of triangles
A
B
C
ABC
A
BC
and
D
E
F
DEF
D
EF
?
2a
1
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In triangle with |AB| > |AC|, find angle BAC
A triangle
A
B
C
ABC
A
BC
with circumcircle
ω
\omega
ω
satisfies
∣
A
B
∣
>
∣
A
C
∣
|AB| > |AC|
∣
A
B
∣
>
∣
A
C
∣
. Points
X
X
X
and
Y
Y
Y
on
ω
\omega
ω
are different from
A
A
A
, such that the line
A
X
AX
A
X
passes through the midpoint of
B
C
BC
BC
,
A
Y
AY
A
Y
is perpendicular to
B
C
BC
BC
, and
X
Y
XY
X
Y
is parallel to
B
C
BC
BC
. Find
∠
B
A
C
\angle BAC
∠
B
A
C
.
1b
1
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Solve n*5^{n-n/p} = p! (p^2+1) + n for prime p and natural n
Find all primes
p
p
p
and positive integers
n
n
n
satisfying
n
⋅
5
n
−
n
/
p
=
p
!
(
p
2
+
1
)
+
n
.
n \cdot 5^{n-n/p} = p! (p^2+1) + n.
n
⋅
5
n
−
n
/
p
=
p
!
(
p
2
+
1
)
+
n
.
1a
1
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2022 + 3^n is a perfect square [2022 Abelkonkurransen Finale, Problem 1a]
Determine all positive integers
n
n
n
such that
2022
+
3
n
2022 + 3^n
2022
+
3
n
is a perfect square.